Renal elimination is the single most important route of elimination of many metabolites and drugs in the body. For patients with decreased renal function, doses need to be decreased to avoid exposure to excessive drug concentrations that may cause toxicity. Renal plasma clearance (CL) measurements are indicated for evaluating renal function, controlling dosing and avoiding toxicity, and for transplant donor and recipient evaluation, amongst other indications.
There are several mechanisms of renal elimination, the first being glomerular filtration. This is a passive mechanism of elimination, whereby ionic substances are renally filtered and excreted. A gold standard test for measuring glomerular filtration rate or plasma clearance is the inulin constant infusion assay. This test, and other constant infusion assays, involves infusion of inulin or other test compound at a constant rate followed by measurement of its concentration in urine and/or plasma over time. Inulin is completely filtered at the glomerulus and is neither secreted nor reabsorbed by the renal tubules. However, inulin constant infusion may not be useful in patients with severely reduced renal function. Moreover, inulin constant infusion testing has been twice reported to cause anaphylactic shock in humans, and other, safer substances and more generally applicable methods are needed for CL estimation. Another gold standard performs a numerical integration of the area under the curve (AUC) for 24 hours or more after bolus (i.e., sudden) injection of a test substance. Gold standard tests have the advantages of accuracy and robustness because they do not require extensive use of curve fitting. Unfortunately, the repeated sampling over extended periods of time makes the numerical integration of AUC impractical, particularly in a clinical setting, and even 24 hour collections may not be long enough to evaluate renal failure
A second mechanism of renal elimination is tubular secretion, which can increase the CL by actively secreting the drug, as opposed to only the passive diffusion in glomerular filtration. The rate of secretion depends on the transporter. Compounds that are secreted usually also undergo glomerular filtration, so renal clearance is the sum of both routes.
A third mechanism affecting the renal clearance of drugs is tubular reabsorption. Some drugs may be reabsorbed after being filtered out of the blood. Thus, the CL may be smaller than expected (when considering only filtration and tubular secretion.) If a drug is “completely” reabsorbed after filtration and no active secretion takes place, the renal clearance will be limited to the amount of drug that leaves the kidney as the urine flows into the bladder. Because of these additional mechanisms, glomerular filtration rate alone using one of the gold standards may not always accurately model CL for some drugs or metabolites.
As a result of the various drawbacks of the gold standard tests, the more common approach to estimating CL is to use curve fitting models. Such models include Sums of Exponential Terms, or SETs, and Gamma Variate (GV). Examples of SETs include ordinary least squares (OLS) regression, Bayesian priors, D-optimal design and Tikhonov regularization (Tk-SET). An example of GV is OLS GV regression. To test how good a curve fit model is, one must test (a) whether one curve fit model is better than another, and (b) whether a curve fit model is good in absolute terms.
The fit of a single exponential term fit to the concentration curve is referred to as an E1 SET model and models that are sums of exponential terms as E2, E3, E4, . . . , En, respectively for 2, 3, 4, . . . , n exponential terms. Current recommendations for assessing renal function or drug elimination after venous bolus injection are to use En>1 for fitting marker concentration curves with 8-13 blood samples. The physical model of linearly coupled, fast-mixing compartments inspired the use of SETs. SETs arise as one of three general solutions for the nth order linear homogeneous ordinary differential equation with constant coefficients.
The use of higher-order SET models leads to a host of problems. This is because when En≧2 SET models are used to model the concentration data they are often not robust enough to converge to statistically acceptable fits. Another problem is accuracy—it is well known that E1 and E2 SET models typically overestimate renal clearance. SET models fail to adequately fit the temporal dependence of marker concentration for radiochelated DTPA. In addition, SET models do not extrapolate properly, and they have small values for the areas under the temporal concentration curve (AUC), which is inappropriate for non-metabolized substances in the case of low (AUC large) to no renal function (AUC→∞).
A variation of the SET method of estimating CL uses numerical integration of area under the curve (AUC) of the concentrations of multiple samples over time and extrapolates the unmeasured area using mono-exponential fits to the last two hours of data. SETs and “AUC plus terminal mono-exponentials” are currently the only bolus models in use for estimating CL. This more complicated augmentation of normal SET models is used most commonly in the case of low renal function. Using mono-exponential extrapolations, it has been estimated that there is a 10% difference between the 4- and 24-hour AUC. It has been assumed that extrapolation using a “terminal” fit with a mono-exponential is less problematic the longer one waits to perform it. However, this observation also suggests that mono-exponential extrapolation consistently underestimates the extrapolated concentration. Another drawback of this method is the relatively long time period over which samples must be collected, causing the AUC method to border on the impractical.
Other curve fit models, gamma variates (GV), have been used to model the temporal dependence of the plasma concentrations of an assortment of drugs, for example, ampicillin, creatinine, chlorpheniramine, chlordiazeproxide, dexamethasone, terbutaline, oxyphenonium bromide, cefroxadin, idopyracet, cefroxadin, T3, pancuronium, inulin, and radiochelated DTPA. GV fits to late samples taken after one hour follow the temporal concentration data well in the fit region. However, direct fitting of a gamma variate function to the temporal concentration data is often ill-conditioned for CL, independent of which samples times are chosen for fitting.
In view of the aforementioned inadequacies of the prior art curve fitting methods, there remains a need to find a CL fit gold standard that agrees with current gold standards such as constant infusion.